Lectures on Gravitation

Contents......Page 10
Preface......Page 8
1.1 Two dimensional geometry......Page 14
1.2 Inertial and gravitational masses......Page 26
1.3 Relativity......Page 29
2.1 Relativistic point particle.......Page 48
2.2 Current and charge densities......Page 59
2.3 Maxwell’s equations in the presence of sources......Page 64
2.4 Motion of a charged particle in EM field......Page 67
2.5 Energy-momentum tensor......Page 69
2.6 Angular momentum.......Page 74
3.1 Principle of equivalence.......Page 78
3.2 Principle of general covariance......Page 83
3.3 Tensor densities......Page 89
4.1 Parallel transport of a vector......Page 96
4.2 Christoffel symbol......Page 109
4.3 Covariant derivative of contravariant tensors......Page 118
4.4 Metric compatibility......Page 124
4.5 Covariant derivative of covariant and mixed tensors......Page 128
4.6 Electromagnetic analogy......Page 131
4.7 Gradient, divergence and curl......Page 133
5.1 Covariant differentiation along a curve......Page 146
5.2 Curvature from derivatives......Page 148
5.3 Parallel transport along a closed curve......Page 152
5.4 Geodesic equation......Page 162
5.5 Derivation of geodesic equation from a Lagrangian......Page 174
6.1 Geodesic as representing gravitational effect......Page 180
6.2 Rotating coordinate system and the Coriolis force......Page 185
6.3 Gravitational red shift......Page 193
6.4 Twin paradox and general covariance......Page 203
6.5 Other equations in the presence of gravitation......Page 206
7.2 Definition of an inertial coordinate frame.......Page 212
7.3 Geodesic deviation......Page 213
7.4 Properties of the curvature tensor......Page 216
7.5 Einstein’s equation......Page 227
7.6 Cosmological constant......Page 235
7.7 Initial value problem......Page 236
7.8 Einstein’s equation from an action......Page 242
8.1 Line element......Page 250
8.2 Connection......Page 253
8.3 Solution of the Einstein equation......Page 257
8.4 Properties of the Schwarzschild solution......Page 270
8.5 Isotropic coordinates......Page 273
9.1 Radar echo experiment......Page 278
9.2 Motion of a particle in a Schwarzschild background......Page 280
9.2.1 Vertical free fall......Page 286
9.2.2 Circular orbit......Page 290
9.3 Motion of light rays in a Schwarzschild background......Page 293
9.4 Perihelion advance of Mercury......Page 300
10 Black holes......Page 308
10.1 Singularities of the metric......Page 309
10.2 Singularities of the Schwarzschild metric......Page 314
10.3 Black holes......Page 319
11.1 Homogeneity and isotropy......Page 324
11.2.1 Close universe......Page 327
11.2.3 Open universe......Page 328
11.3 Hubble’s law......Page 329
11.4 Evolution equation......Page 333
11.4.1 k = 1......Page 338
11.4.2 k = 0......Page 340
11.4.3 k = -1......Page 341
11.5 Big bang theory and blackbody radiation......Page 342
Index......Page 348